DATA 2020

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day = read.csv("/Users/Murphy1/Desktop/DATA2020/Project2020/data/hour.csv", header = TRUE)
head(day)

library(dplyr)

#select variables season, workingday, weathersit, temp, atemp, hum, windspeed
day1 <- dplyr::select(day, -c(instant, dteday))
head(day1)

#drop atemp
day1 <- dplyr::select(day1, -c(atemp))
day1

# Check Line

fit.all = lm(cnt ~ season + yr + mnth + hr + holiday + workingday + weathersit + temp + hum + windspeed, data = day1)

intercept_only <- lm(cnt ~ 1, data=day1)

forward <- step(intercept_only, direction='forward', 
                scope=formula(fit.all))

Start:  AIC=180764.7
cnt ~ 1

             Df Sum of Sq       RSS    AIC
+ temp        1  93677759 478083832 177657
+ hr          1  88790198 482971393 177834
+ hum         1  59618351 512143240 178853
+ yr          1  35876722 535884870 179641
+ season      1  18127040 553634551 180207
+ weathersit  1  11598301 560163290 180411
+ mnth        1   8321115 563440476 180512
+ windspeed   1   4970060 566791531 180615
+ holiday     1    546889 571214702 180750
+ workingday  1    524387 571237204 180751
<none>                    571761591 180765

Step:  AIC=177657
cnt ~ temp

             Df Sum of Sq       RSS    AIC
+ hr          1  66728221 411355610 175046
+ hum         1  49874585 428209247 175744
+ yr          1  31342263 446741569 176481
+ windspeed   1   6021342 472062489 177439
+ weathersit  1   5880680 472203151 177444
+ season      1   1696802 476387030 177597
+ mnth        1    906463 477177369 177626
+ holiday     1    225697 477858135 177651
<none>                    478083832 177657
+ workingday  1     35467 478048365 177658

Step:  AIC=175046.4
cnt ~ temp + hr

             Df Sum of Sq       RSS    AIC
+ yr          1  32229070 379126540 173631
+ hum         1  25434149 385921461 173939
+ weathersit  1   5638993 405716617 174809
+ season      1   2995571 408360039 174921
+ windspeed   1   1712372 409643238 174976
+ mnth        1   1525503 409830107 174984
+ holiday     1    260174 411095436 175037
+ workingday  1     54016 411301595 175046
<none>                    411355610 175046

Step:  AIC=173630.5
cnt ~ temp + hr + yr

             Df Sum of Sq       RSS    AIC
+ hum         1  20865128 358261412 172649
+ weathersit  1   5240157 373886383 173391
+ season      1   3515639 375610901 173471
+ mnth        1   1811568 377314972 173549
+ windspeed   1   1810495 377316045 173549
+ holiday     1    307712 378818828 173618
+ workingday  1     66617 379059923 173629
<none>                    379126540 173631

Step:  AIC=172648.7
cnt ~ temp + hr + yr + hum

             Df Sum of Sq       RSS    AIC
+ season      1   7325038 350936374 172292
+ mnth        1   4840141 353421271 172414
+ holiday     1    367007 357894405 172633
+ weathersit  1    133898 358127514 172644
+ workingday  1    117604 358143808 172645
<none>                    358261412 172649
+ windspeed   1     15517 358245895 172650

Step:  AIC=172291.7
cnt ~ temp + hr + yr + hum + season

             Df Sum of Sq       RSS    AIC
+ holiday     1    371167 350565206 172275
+ windspeed   1    165365 350771008 172286
+ workingday  1    131868 350804506 172287
<none>                    350936374 172292
+ weathersit  1     37972 350898402 172292
+ mnth        1       799 350935575 172294

Step:  AIC=172275.3
cnt ~ temp + hr + yr + hum + season + holiday

             Df Sum of Sq       RSS    AIC
+ windspeed   1    165343 350399863 172269
+ workingday  1     47102 350518104 172275
+ weathersit  1     42022 350523184 172275
<none>                    350565206 172275
+ mnth        1         0 350565206 172277

Step:  AIC=172269.1
cnt ~ temp + hr + yr + hum + season + holiday + windspeed

             Df Sum of Sq       RSS    AIC
+ weathersit  1     73738 350326125 172267
+ workingday  1     48197 350351667 172269
<none>                    350399863 172269
+ mnth        1         1 350399863 172271

Step:  AIC=172267.5
cnt ~ temp + hr + yr + hum + season + holiday + windspeed + weathersit

             Df Sum of Sq       RSS    AIC
+ workingday  1     53900 350272225 172267
<none>                    350326125 172267
+ mnth        1         1 350326124 172269

Step:  AIC=172266.8
cnt ~ temp + hr + yr + hum + season + holiday + windspeed + weathersit + 
    workingday

       Df Sum of Sq       RSS    AIC
<none>              350272225 172267
+ mnth  1      1.92 350272223 172269

summary(forward)

Call:
lm(formula = cnt ~ temp + hr + yr + hum + season + holiday + 
    windspeed + weathersit + workingday, data = day1)

Residuals:
    Min      1Q  Median      3Q     Max 
-366.86  -93.52  -27.97   60.40  644.46 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)   -9.4775     6.5712  -1.442 0.149246    
temp         282.8600     6.0073  47.086  < 2e-16 ***
hr             7.6962     0.1649  46.680  < 2e-16 ***
yr            80.9848     2.1668  37.376  < 2e-16 ***
hum         -197.1206     6.8663 -28.708  < 2e-16 ***
season        20.0853     1.0486  19.154  < 2e-16 ***
holiday      -25.1349     6.6610  -3.773 0.000162 ***
windspeed     29.5888     9.4015   3.147 0.001651 ** 
weathersit    -3.7797     1.9044  -1.985 0.047187 *  
workingday     3.9204     2.3980   1.635 0.102096    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 142 on 17369 degrees of freedom
Multiple R-squared:  0.3874,    Adjusted R-squared:  0.3871 
F-statistic:  1220 on 9 and 17369 DF,  p-value: < 2.2e-16

#forward

backward <- step(fit.all, direction='backward')

Start:  AIC=172268.8
cnt ~ season + yr + mnth + hr + holiday + workingday + weathersit + 
    temp + hum + windspeed

             Df Sum of Sq       RSS    AIC
- mnth        1         2 350272225 172267
<none>                    350272223 172269
- workingday  1     53901 350326124 172269
- weathersit  1     79432 350351655 172271
- windspeed   1    199754 350471977 172277
- holiday     1    286716 350558939 172281
- season      1   2453586 352725809 172388
- hum         1  16562243 366834466 173070
- yr          1  28170183 378442406 173611
- hr          1  43896453 394168676 174319
- temp        1  44245313 394517536 174334

Step:  AIC=172266.8
cnt ~ season + yr + hr + holiday + workingday + weathersit + 
    temp + hum + windspeed

             Df Sum of Sq       RSS    AIC
<none>                    350272225 172267
- workingday  1     53900 350326125 172267
- weathersit  1     79442 350351667 172269
- windspeed   1    199752 350471977 172275
- holiday     1    287145 350559370 172279
- season      1   7398820 357671045 172628
- hum         1  16620569 366892794 173070
- yr          1  28171966 378444191 173609
- hr          1  43943958 394216183 174319
- temp        1  44711469 394983694 174353

summary(backward)

Call:
lm(formula = cnt ~ season + yr + hr + holiday + workingday + 
    weathersit + temp + hum + windspeed, data = day1)

Residuals:
    Min      1Q  Median      3Q     Max 
-366.86  -93.52  -27.97   60.40  644.46 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)   -9.4775     6.5712  -1.442 0.149246    
season        20.0853     1.0486  19.154  < 2e-16 ***
yr            80.9848     2.1668  37.376  < 2e-16 ***
hr             7.6962     0.1649  46.680  < 2e-16 ***
holiday      -25.1349     6.6610  -3.773 0.000162 ***
workingday     3.9204     2.3980   1.635 0.102096    
weathersit    -3.7797     1.9044  -1.985 0.047187 *  
temp         282.8600     6.0073  47.086  < 2e-16 ***
hum         -197.1206     6.8663 -28.708  < 2e-16 ***
windspeed     29.5888     9.4015   3.147 0.001651 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 142 on 17369 degrees of freedom
Multiple R-squared:  0.3874,    Adjusted R-squared:  0.3871 
F-statistic:  1220 on 9 and 17369 DF,  p-value: < 2.2e-16

#day1$cnt <- log(day1$cnt)
mod_forward <- lm(formula = cnt ~ temp + hr + yr + hum + season + holiday + 
    windspeed + weathersit + workingday, data = day1)
plot(mod_forward, which = c(1,2,3,4))

mod_backward <- lm(formula = cnt ~ season + yr + hr + holiday + workingday + 
    weathersit + temp + hum + windspeed, data = day1)
plot(mod_backward, which = c(1,2,3,4))

day1$cnt <- log(day1$cnt)
mod_backward <- lm(formula = cnt ~ season + yr + hr + holiday + workingday + 
    weathersit + temp + hum + windspeed, data = day1)
plot(mod_backward, which = c(1,2,3,4))

# Check the variance 
library(car)
ncvTest(fit.all)

Non-constant Variance Score Test 
Variance formula: ~ fitted.values 
Chisquare = 2063.008, Df = 1, p = < 2.22e-16

backward <- step(fit.all, direction='backward')

Start:  AIC=172268.8
cnt ~ season + yr + mnth + hr + holiday + workingday + weathersit + 
    temp + hum + windspeed

             Df Sum of Sq       RSS    AIC
- mnth        1         2 350272225 172267
<none>                    350272223 172269
- workingday  1     53901 350326124 172269
- weathersit  1     79432 350351655 172271
- windspeed   1    199754 350471977 172277
- holiday     1    286716 350558939 172281
- season      1   2453586 352725809 172388
- hum         1  16562243 366834466 173070
- yr          1  28170183 378442406 173611
- hr          1  43896453 394168676 174319
- temp        1  44245313 394517536 174334

Step:  AIC=2483.88
cnt ~ season + yr + hr + holiday + workingday + weathersit + 
    temp + hum + windspeed

             Df Sum of Sq   RSS    AIC
- weathersit  1       0.0 20026 2481.9
<none>                    20026 2483.9
- workingday  1      12.4 20038 2492.6
- windspeed   1      13.0 20039 2493.2
- holiday     1      21.5 20048 2500.6
- season      1     477.2 20503 2891.2
- yr          1     773.6 20800 3140.6
- hum         1     881.3 20907 3230.3
- temp        1    2336.1 22362 4399.4
- hr          1    7693.2 27719 8131.6

Step:  AIC=2481.9
cnt ~ season + yr + hr + holiday + workingday + temp + hum + 
    windspeed

             Df Sum of Sq   RSS    AIC
<none>                    20026 2481.9
- workingday  1      12.5 20039 2490.7
- windspeed   1      13.2 20039 2491.4
- holiday     1      21.5 20048 2498.6
- season      1     478.6 20505 2890.3
- yr          1     774.0 20800 3139.0
- hum         1    1114.1 21140 3420.8
- temp        1    2349.3 22376 4407.7
- hr          1    7788.8 27815 8189.5

summary(backward)

Call:
lm(formula = cnt ~ season + yr + hr + holiday + workingday + 
    temp + hum + windspeed, data = day1)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.9871 -0.6398  0.1151  0.7121  3.1887 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  2.631172   0.049540  53.112  < 2e-16 ***
season       0.161355   0.007920  20.374  < 2e-16 ***
yr           0.424309   0.016376  25.911  < 2e-16 ***
hr           0.101812   0.001239  82.193  < 2e-16 ***
holiday     -0.217687   0.050364  -4.322 1.55e-05 ***
workingday  -0.059567   0.018112  -3.289 0.001008 ** 
temp         2.045017   0.045303  45.141  < 2e-16 ***
hum         -1.438355   0.046271 -31.086  < 2e-16 ***
windspeed    0.237638   0.070215   3.384 0.000715 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.074 on 17370 degrees of freedom
Multiple R-squared:  0.4782,    Adjusted R-squared:  0.478 
F-statistic:  1990 on 8 and 17370 DF,  p-value: < 2.2e-16

library(plyr)
library(readr)
library(dplyr)
library(glmnet)
library(tibble) 
library(caret)

set.seed(42)
train_index <- createDataPartition(day1$cnt, p = 0.8, list = FALSE)
train_set <- day1[train_index,]
test_set <- day1[-train_index,]

# Train the multiple linear regression model
model <- lm(formula = cnt ~ season + yr + hr + holiday + workingday + 
    weathersit + temp + hum + windspeed, data = train_set)

# Make predictions on the test set
predictions <- predict(model, newdata = test_set)

# Evaluate the model
mae <- mean(abs(test_set$cnt - predictions))
mse <- mean((test_set$cnt - predictions)^2)
r_squared <- 1 - (sum((test_set$cnt - predictions)^2) / sum((test_set$cnt - mean(test_set$cnt))^2))

cat('Mean Absolute Error:', mae, '\n')

Mean Absolute Error: 0.8430607 

cat('Mean Squared Error:', mse, '\n')

Mean Squared Error: 1.156954 

cat('R-squared:', r_squared, '\n')

R-squared: 0.4713828 

rmse <- function(actual, predicted) {
  return(sqrt(mean((actual - predicted)^2)))
}
rmse_value <- rmse(test_set$cnt, predictions)
cat('Root Mean Squared Error:', rmse_value, '\n')

Root Mean Squared Error: 1.075618 

day2 <- day1[ , -which(names(day1) == "cnt")]
pca <- prcomp(day2,scale=TRUE)
variance <- summary(pca)$importance[2,]
df <- data.frame(Component = 1:length(variance), Variance = variance)
df_top_10 <- df[1:10,]
df_top_10 <- df_top_10[order(-df_top_10$Variance),]
df_top_10

pc <- pca$rotation[, 1: 10]
z <- melt(pc)
colnames(z) <- c("Variable", "PC", "value")
plot <- ggplot(z, aes(x = PC, y = Variable, fill= value)) +
 geom_tile(color = 'white')+
 scale_fill_gradient2(low = 'brown', high = 'wheat', mid = 'ivory',
   midpoint = 0, limit = c(-1,1), space = 'Lab')
plot

pca_data <- as.data.frame(predict(pca))
df2 <- pca_data[, 1:2]
df1 <- subset(day1, select = cnt)
combined_df <- cbind(df1, df2)
library(caret)
train_index <- createDataPartition(combined_df$cnt, p = 0.8, list = FALSE)
train_set <- combined_df[train_index,]
test_set <- combined_df[-train_index,]
# Train the multiple linear regression model
model2 <- lm(cnt ~ ., data = train_set)
summary(model2)

Call:
lm(formula = cnt ~ ., data = train_set)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.9418 -0.4606  0.1181  0.6162  2.7952 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 4.537284   0.007966  569.55   <2e-16 ***
PC1         0.720840   0.005032  143.24   <2e-16 ***
PC2         0.113545   0.005553   20.45   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.9394 on 13901 degrees of freedom
Multiple R-squared:  0.601, Adjusted R-squared:  0.601 
F-statistic: 1.047e+04 on 2 and 13901 DF,  p-value: < 2.2e-16

# Make predictions on the test set
predictions <- predict(model2, newdata = test_set)
# Evaluate the model
mae <- mean(abs(test_set$cnt - predictions))
mse <- mean((test_set$cnt - predictions)^2)
r_squared <- 1 - (sum((test_set$cnt - predictions)^2) / sum((test_set$cnt - mean(test_set$cnt))^2))
summary(predictions)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  1.916   3.689   4.405   4.555   5.292   8.573